Theorems · Theorem · commutative algebra

Submodule.eq_smul_of_le_smul_of_le_jacobson

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {I J : Ideal R}
  {N : Submodule R M}, N.FG → N ≤ I • N → I ≤ J.jacobson → N = J • N

Nakayama's Lemma - A slightly more general version of (2) in [Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV). See also eq_bot_of_le_smul_of_le_jacobson_bot for the special case when J = ⊥.

Defined in

Mathlib.RingTheory.Nakayama

Cited by

3 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule